Properties of Triangles 1 Question 5
5. If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A$ is
(2010)
(a) $\frac{1}{2}$
(b) $\frac{\sqrt{3}}{2}$
(c) 1
(d) $\sqrt{3}$
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Answer:
Correct Answer: 5. (d)
Solution:
- Since, $A, B, C$ are in AP.
$\Rightarrow \quad 2 B=A+C$ i.e. $\angle B=60^{\circ}$
$\therefore \quad \frac{a}{c}(2 \sin C \cos C)+\frac{c}{a}(2 \sin A \cos A)$
$$ =2 k(a \cos C+c \cos A) $$
$$ \text { using, } \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{k} $$
$=2 k(b)$
$=2 \sin B \quad$ [using $b=a \cos C+c \cos A$ ] $=\sqrt{3}$
